Hokkaido Mathematical Journal

CHO Yonggeun, LEE Sanghyuk, SHIM Yongsun,

A maximal inequality associated to Schr\"{o}dinger type equation.

Hokkaido Mathematical Journal, 35 (2006) pp.767-778

Fulltext

PDF

Abstract

In this note, we consider a maximal operator $\sup_{t \in \mathbb{R}}|u(x,t)| = \sup_{t \in \mathbb{R}}|e^{it\Omega(D)}f(x)|$, where $u$ is the solution to the initial value problem $u_t = i\Omega(D)u$, $u(0) = f$ for a $C^2$ function $\Omega$ with some growth rate at infinity. We prove that the operator $\sup_{t \in \mathbb{R}}|u(x,t)|$ has a mapping property from a fractional Sobolev space $H^\fraca{1}{4}$ with additional angular regularity in which the data lives to $L^2((1 + |x|)^{-b}dx) (b > 1)$ . This mapping property implies the almost everywhere convergence of $u(x,t)$ to $f$ as $t \to 0$, if the data $f$ has an angular regularity as well as $H^\frac{1}{4}$ regularity.

MSC(Primary)42A45
MSC(Secondary)42B25
Uncontrolled KeywordsSchr\"{o}dinger type equation, maximal operator, angular regularity